Last visit was: 23 May 2024, 03:01 It is currently 23 May 2024, 03:01
Decision Tracker
My Rewards
Unanswered
Active Topics
New posts
New comers' posts
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel

What is the units digit of 66^6 - 5^55?

SORT BY:
Date
Kudos
Tags:
Find Similar Topics 
Show Tags
Hide Tags
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 93417
Own Kudos [?]: 625983 [96]
Given Kudos: 81940
Send PM
CAT Tests Forum Quiz Mode
What is the units digit of 66^6 - 5^55? [#permalink] What is the units digit of 66^6 - 5^55?   10 Oct 2017, 01:54
3
Kudos
93
Bookmarks
Expert Reply
00:00
A
B
C
D
E

Difficulty:

85% (hard)

Question Stats:

27% (00:59) correct 73% (00:46) wrong based on 801 sessions
Hide Show timer Statistics
What is the units digit of 66^6 - 5^55?

A. 0
B. 1
C. 5
D. 6
E. 9

Experience a GMAT Club Test Questions
Yes, you've landed on a GMAT Club Tests question
Craving more? Unlock our full suite of GMAT Club Tests here
Want to experience more? Get a taste of our tests with our free trial today
Rise to the challenge with GMAT Club Tests. Happy practicing!
GMAT Club Tests Check Our Products Try Free Tests
­
ShowHide Answer
Official Answer
E
Most Helpful Reply
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 93417
Own Kudos [?]: 625983 [17]
Given Kudos: 81940
Send PM
CAT Tests Forum Quiz Mode
Re: What is the units digit of 66^6 - 5^55? [#permalink] What is the units digit of 66^6 - 5^55?   03 Dec 2019, 00:50
6
Kudos
11
Bookmarks
Expert Reply
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:



What is the units digit of 66^6 - 5^55?

A. 0
B. 1
C. 5
D. 6
E. 9


Official Solution:

A number with the units digit of 6, in any positive integer power, will have 6 as its units digit. So, the units digit of \(66^6\) is 6.

A number with the units digit of 5, in any positive integer power, will have 5 as its units digit. So, the units digit of \(5^{55}\) is 5.

Here, you have to be careful and not assume that the units digit of \(66^6 - 5^{55}\) will be \(6 - 5 = 1\) because you should notice that \(5^{55}\) is MUCH larger number than \(66^6\). Why? \(5^{55}=5^{5*11}=(5^5)^{11}\). Now, \(5^5>66\) and \(11>6\), thus \((5^5)^{11}>66^6\).

Therefore, the units digit of \(66^6 - 5^{55}\) will be \(\{ smaller \ number \ with \ the \ units \ digit \ of \ 6 \} - \{ larger \ number \ with \ the \ units \ digit \ of \ 5\} = 9\). For example, \(16-25=-9\)


Answer: E
General Discussion
User avatar
L
NandishSS
Director
Director
Joined: 06 Jan 2015
Posts: 737
Own Kudos [?]: 1591 [4]
Given Kudos: 579
Location: India
Concentration: Operations, Finance
GPA: 3.35
WE:Information Technology (Computer Software)
Send PM
Re: What is the units digit of 66^6 - 5^55? [#permalink] What is the units digit of 66^6 - 5^55?   Updated on: 10 Oct 2017, 19:59
1
Kudos
3
Bookmarks
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:



What is the units digit of 66^6 - 5^55?

A. 0
B. 1
C. 5
D. 6
E. 9


The cycle of 5 and 6, when raised to any power, will be same respectively.

For ex 5^1=5

5^2=25

5^3=125

And 6^1=6

6^2=36

6^3=216

However 66^6 < 5^55 unit digit would be negative

So 6-25 =-9

Ans :
Show Spoiler
E

Originally posted by NandishSS on 10 Oct 2017, 02:03.
Last edited by NandishSS on 10 Oct 2017, 19:59, edited 1 time in total.
User avatar
D
shashankism
Director
Director
Joined: 13 Mar 2017
Affiliations: IIT Dhanbad
Posts: 625
Own Kudos [?]: 593 [2]
Given Kudos: 88
Location: India
Concentration: General Management, Entrepreneurship
GPA: 3.8
WE:Engineering (Energy and Utilities)
Send PM
Re: What is the units digit of 66^6 - 5^55? [#permalink] What is the units digit of 66^6 - 5^55?   10 Oct 2017, 02:32
2
Kudos
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:



What is the units digit of 66^6 - 5^55?

A. 0
B. 1
C. 5
D. 6
E. 9


Since 6^2 = 36, 6^3 = 216 .. So unit's digit of 66^6 will be 6
Also 5^2 = 25. 5^3 = 125 , 5^4 = 625 .. So unit's digit of 5^55 will be 5

So, units digit of 66^6 - 5^55 = 6-5 =1

Answer B
User avatar
P
Mahmud6
Retired Moderator
Joined: 10 Mar 2013
Status:The best is yet to come.....
Posts: 397
Own Kudos [?]: 833 [1]
Given Kudos: 235
Send PM
GMAT ToolKit User
Re: What is the units digit of 66^6 - 5^55? [#permalink] What is the units digit of 66^6 - 5^55?   10 Oct 2017, 05:39
1
Bookmarks
This problem is dealing with the last digit of a power. Details are described in the following link https://gmatclub.com/forum/math-number- ... 88376.html.

Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base. So, the last digit of \(66^6\) is 6 and the last digit of \(5^{55}\) is 5. Therefore, the answer would be xxxx6-xxxx5=1, B.
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 93417
Own Kudos [?]: 625983 [1]
Given Kudos: 81940
Send PM
CAT Tests Forum Quiz Mode
Re: What is the units digit of 66^6 - 5^55? [#permalink] What is the units digit of 66^6 - 5^55?   10 Oct 2017, 05:45
1
Kudos
Expert Reply
NandishSS wrote:

The cycle of 5 and 6, when raised to any power, will be same respectively.

For ex 5^1=5

5^2=25

5^3=125

And 6^1=6

6^2=36

6^3=216

So 6-5 = 1

Ans :
Show Spoiler
B


shashankism wrote:
Since 6^2 = 36, 6^3 = 216 .. So unit's digit of 66^6 will be 6
Also 5^2 = 25. 5^3 = 125 , 5^4 = 625 .. So unit's digit of 5^55 will be 5

So, units digit of 66^6 - 5^55 = 6-5 =1

Answer B


Mahmud6 wrote:
This problem is dealing with the last digit of a power. Details are described in the following link https://gmatclub.com/forum/math-number- ... 88376.html.

Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base. So, the last digit of \(66^6\) is 6 and the last digit of \(5^{55}\) is 5. Therefore, the answer would be xxxx6-xxxx5=1, B.



Simple tricks are the best... The answer is
Show Spoiler
not B
avatar
B
mj12345
Intern
Intern
Joined: 02 Feb 2016
Posts: 27
Own Kudos [?]: 18 [4]
Given Kudos: 134
Location: United States
Schools: McCombs '21 (WD) Haas '21 (D) Duke '21 (D) Darden '21 (S) Kellogg '21 (S) Wharton '21 (D) Tuck '21 (D) Booth '21 ISB '20 (WL)
GMAT 1: 710 Q49 V38
GPA: 3.5
Send PM
Reviews Badge
Re: What is the units digit of 66^6 - 5^55? [#permalink] What is the units digit of 66^6 - 5^55?   10 Oct 2017, 05:55
1
Kudos
3
Bookmarks
The last digit will be 1 if 66^6 > 5^55.

I don’t know the best way to explain this; but looking at the numbers 5^55 >>> 66^6. So in this case:

xxx6 - xxxxxxx5 = -yyyyy9

The units digit will be 9.


Sent from my iPhone using GMAT Club Forum
User avatar
L
NandishSS
Director
Director
Joined: 06 Jan 2015
Posts: 737
Own Kudos [?]: 1591 [1]
Given Kudos: 579
Location: India
Concentration: Operations, Finance
GPA: 3.35
WE:Information Technology (Computer Software)
Send PM
Re: What is the units digit of 66^6 - 5^55? [#permalink] What is the units digit of 66^6 - 5^55?   10 Oct 2017, 06:41
Bunuel Trapped !!! mykbdj Caught It correctly :-) Agree with his explaination :-)
User avatar
B
atulindia
Intern
Intern
Joined: 08 Mar 2017
Posts: 47
Own Kudos [?]: 27 [0]
Given Kudos: 28
Location: India
Concentration: Operations, General Management
GPA: 2.79
WE:Project Management (Manufacturing)
Send PM
Reviews Badge
Re: What is the units digit of 66^6 - 5^55? [#permalink] What is the units digit of 66^6 - 5^55?   10 Oct 2017, 08:50
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:



What is the units digit of 66^6 - 5^55?

A. 0
B. 1
C. 5
D. 6
E. 9



Unit digit of 66^6 is 6 and 5^55 is 5
But we do not know which one is greater, so unit of difference can be 1 or 9
But from given data 66x66x66x66x66x66 or 66^6 <<< 125x125x125x125x125x125 x 5^37 or 5^55
Hence unit digit is 9 and answer is E
User avatar
B
aashaybaindurgmat
Intern
Intern
Joined: 08 Jun 2017
Posts: 9
Own Kudos [?]: 52 [1]
Given Kudos: 4
Send PM
Re: What is the units digit of 66^6 - 5^55? [#permalink] What is the units digit of 66^6 - 5^55?   14 Oct 2017, 22:42
1
Bookmarks
Answer : 9
Answer : E

Obviously 66^6 will give a units digit of 6.
and 5^55 will give a units digit of 5.

But 6-5 = 1 will only apply if there is no carrying forward meaning, that the "6" number is larger than the "5" Number.

Doing some math though tells us that the "5" number is indeed much larger, which means carrying forward does take place.
Making the subtraction look much more like 15 - 6 = 9, than the aforementioned 6 - 5 = 1.

Therefore the answer is E.
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 93417
Own Kudos [?]: 625983 [0]
Given Kudos: 81940
Send PM
CAT Tests Forum Quiz Mode
Re: What is the units digit of 66^6 - 5^55? [#permalink] What is the units digit of 66^6 - 5^55?   24 Dec 2018, 04:55
Expert Reply
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:



What is the units digit of 66^6 - 5^55?

A. 0
B. 1
C. 5
D. 6
E. 9


Par of GMAT CLUB'S New Year's Quantitative Challenge Set

User avatar
D
BrushMyQuant
Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1779
Own Kudos [?]: 2106 [1]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
Send PM
Re: What is the units digit of 66^6 - 5^55? [#permalink] What is the units digit of 66^6 - 5^55?   15 Dec 2023, 06:42
1
Kudos
Expert Reply
Top Contributor
We need to find the units digit of 66^6 - 5^55?

Both 6 and 5 have a cycle of 1 for units digit

Watch these videos to Master Cyclicity for Unit's Digit

That means any positive power of 5 and 6 will have units digit of 5 and 6 respectively

Example:
Units digit of \(5^1\) = 5
Units digit of \(5^2\) = 5
Units digit of \(5^3\) = 5

Units digit of \(6^1\) = 6
Units digit of \(6^2\) = 6
Units digit of \(6^3\) = 6

But \(5^{55} > 6^6\)
=> Units digit of 6\(6^6 - 5^{55}\) = -X5 + 6 = -9 (Similar to 6 - 15)

So, Answer will be E
Hope it helps!
GMAT Club Bot
gmatclubot
Re: What is the units digit of 66^6 - 5^55? [#permalink]
15 Dec 2023, 06:42
Moderator:
Bunuel
Math Expert
93417 posts